Maximal-abelianity conjecture for the Galois symmetry group of SIC-POVMs

Let $S$ be the symmetry group of the overlap map of a fiducial, let $C(S)$ be its centralizer in $\mathrm{GL}_2(\mathbb{Z}_{\bar d})$, and let $M$ be a subgroup of $C(S)$ such that the associated Galois group is isomorphic to $M/S$. Maximal-abelianity conjecture. The subgroup $M$ is a maximal abelian subgroup of $C(S)$. All known solutions in the source satisfy this property, but the source does not establish it in general.